Because ENL is in its infancy, all the graphs shown here express broad conceptual relationships using notional curves.

When used in the physical or social sciences, graphs fall into two categories: those that express known quantitative relationships through empirical curves, and those that express general conceptual relationships through notional curves. Each category is indispensable in its proper sphere.

So long as the underlying ideas are sound, notional curves can accurately depict general relationships, permitting both the analyst and the public to draw broad conclusions. Such curves are frequently indispensable in the early stages of theoretical development.

The test of these graphs is not whether the quantities have been measured, but whether they are in principle measurable—that is, if they are based on well-founded concepts and coherent reasoning.

Following is an example of each from scientific sources.

A standard physics textbook shows a graph that relates the force on a car as a function of time during a crash impact. Force is on the vertical axis, in Newtons. Time is on the horizontal axis, in seconds. The graph’s curve, which traces the destruction of a Mercedes-Benz over 120 milliseconds in a test environment, is based on actual measurements and is therefore empirical.1

A completely different situation exists for scientists who are studying the prevalence of life in the universe. Controlled conditions do not exist in this domain, so direct measurements are impossible.

One intriguing book on the subject shows a graph that relates species diversity to mass extinction events. Diversity is on the vertical axis, without units. Mass extinction events are on the horizontal axis, also without units. There are two curves, depicting two possible relationships between the variables. Although these curves are entirely notional, they express the authors' key ideas succinctly and accurately.2

Without notional graphs, the authors would have been unable to posit these hypothetical relationships, thereby undercutting the scientific development of their field. More generally, such a restriction would negate the development of any new field or framework, which must necessarily begin with raw ideas and provisional relationships.3

A good example of such a relationship in economics is the Laffer curve, which relates tax rates to tax revenues. According to legend, the conservative economist Arthur Laffer graphed this relationship on a napkin in a Washington restaurant in 1981.

The curve, which is still being printed in economics texts, shows tax revenues initially rising as the tax rate increases, and then declining once the tax rate reaches a threshold point. There are no units on either axis, and at the time Laffer doodled on his napkin, there was little or no data to support the curve's shape.

The Laffer curve is thus entirely notional, but it nevertheless played a major role in the upsurge of supply-side economics in the 1980s.4

Notional curves are extremely important because they provide investigators with both the guidance and the incentive to perform the necessary research to transform them into empirical curves.

Once a curve is drawn, economists can examine statistical data and try to substantiate the relationship being depicted. Where such information is lacking, they can initiate efforts to collect it.

2. Peter D. Ward and Donald Brownlee, Rare Earth: Why Complex Life Is Uncommon in the Universe (New York: Springer-Verlag, 2000), 172.
Another excellent example of the ￼notional-before-empirical order is provided by physicist Freeman Dyson, in his introduction to fellow physicist Richard Feynman's book, The Pleasure of Finding Things Out (Cambridge, Massachusetts: Perseus Books, 1999). Dyson writes: "For years I watched as Feynman perfected his way of describing nature with pictures and diagrams, until he had tied down the loose ends and removed the inconsistencies. Then he began to calculate numbers, using his diagrams as a guide." (p. ix.) These pictures, now known as Feynman Diagrams, are indispensable tools for describing the behavior of subatomic particles.

3. This point was addressed by John Kenneth Galbraith: "…precision in scholarly discourse not only serves as an aid in the communication of ideas, but it acts to eliminate unwelcome currents of thought, for these can almost invariably be dismissed as imprecise." (The Affluent Society, 1958, p. 270).

4. For an example of the Laffer curve, see Lipsey and Ragan, Economics (Toronto: Addison Wesley Longman, 2001), 432.
A different version can be found in Michael Parkin, Economics (Reading: Addison-Wesley Publishing Company, 1990), 909. What is amusing in these depictions is that the two sets of authors differ on the orientation of the axes. Lipsey and Ragan comply with mathematical convention by placing the tax rate (the independent variable) on the X-axis. Parkin, however, uses the supply-demand curve convention by placing tax rate on the Y-axis.
Standard economists, having for some reason decided to invert mathematical convention for supply-demand graphs, are evidently unable to decide among themselves whether this inversion should apply to all their graphs. This issue is addressed further in the treatment of potential value.